A timer doesn't work because it takes longer to boil three cups of water compared to one. The bimetallic mechanism is agnostic of the amount of rice being cooked.
In the design being described there is an outer water reservoir where the water is being boiled away specifically as a timer. An alternative would be to have a bimetallic strip detect when the rice pot comes to temperature, lower the temp, and start a mechanical timer that could be set by the user. The advantage would be a significant energy savings vs. boiling water away as a kind of crude timer.